Rogers-Selberg Identities

Author

Eric W. Weisstein
December 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Rogers-SelbergIdentities.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rogers-SelbergIdentities.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identities

A(q)=(∑_(n=0)^∞)((q^(2(n^2)))/((Subscript[((q^2);(q^2)), n])(Subscript[(-q;q), 2n])))=((Subscript[((q^3),(q^4),(q^7);(q^7)), ∞])/(Subscript[((q^2);(q^2)), ∞]))

Sum[((q^(2(n^2)))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n])),{n,0,∞}]

Series[Sum[((q^(2(n^2)))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n])),{n,0,20}],{q,0,20}]

Series[((Times@@QPochhammer[{(q^3),(q^4),(q^7)},(q^7)])/(QPochhammer[(q^2),(q^2)])),{q,0,20}]

CoefficientList[%,q]

B(q)=(∑_(n=0)^∞)((q^(2(n^2)+2n))/((Subscript[((q^2);(q^2)), n])(Subscript[(-q;q), 2n])))=((Subscript[((q^2),(q^5),(q^7);(q^7)), ∞])/(Subscript[((q^2);(q^2)), ∞]))

Sum[((q^(2(n^2)+2n))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n])),{n,0,∞}]//Timing

Series[Sum[((q^(2(n^2)+2n))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n])),{n,0,20}],{q,0,20}]

Series[((Times@@QPochhammer[{(q^2),(q^5),(q^7)},(q^7)])/(QPochhammer[(q^2),(q^2)])),{q,0,20}]

CoefficientList[%,q]

C(q)=(∑_(n=0)^∞)((q^(2(n^2)+2n))/((Subscript[((q^2);(q^2)), n])(Subscript[(-q;q), 2n+1])))=((Subscript[(q,(q^6),(q^7);(q^7)), ∞])/(Subscript[((q^2);(q^2)), ∞]))

Sum[((q^(2(n^2)+2n))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n+1])),{n,0,∞}]//Timing

Series[Sum[((q^(2(n^2)+2n))/(QPochhammer[(q^2),(q^2),n]QPochhammer[-q,q,2n+1])),{n,0,20}],{q,0,20}]

Series[((Times@@QPochhammer[{q,(q^6),(q^7)},(q^7)])/(QPochhammer[(q^2),(q^2)])),{q,0,20}]

CoefficientList[%,q]